First we decompose each \(p_\lambda \)  in (REF ) into the sum of a smooth concave term and a simple \(\ell _1\)  penalty term, of the form \(p_\lambda (|x|) = {\widetilde{p}}_\lambda (|\beta |) + \lambda |\beta |\) ,
where \({\widetilde{p}}_{\lambda }\)  is a Lipschitz-smooth concave function [1]}. The goal is to treat the smooth component as part of the likelihood, leaving only a simpler \(\ell _1\)  penalty. This decomposition can be done to both the parameterwise penalties \(p_{\lambda _1}\)  and the fusion penalties \(p_{\lambda _2}\) , rewriting (REF ) as
\(P_{\lambda }(\psi ) = \sum _{g=1}^3 \sum _{j=1}^{d_{g}}{\widetilde{p}}_{\lambda _1}(|\beta _{gj}|) + \sum _{g\ne g^{\prime }}\sum _{j=1}^{d_g} {\widetilde{p}}_{\lambda _2}\left(|\beta _{gj}-\beta _{g^{\prime }j}|\right) + \lambda _{1}\Vert \beta \Vert _{1} + \Omega _{\lambda _2}(\psi ),\) 
